Chapter 5 Inverse Functions and Applications Section 5.1

Section 5.1 Finding the Inverse: Numerical, Graphical, and Symbolical Approaches Definition of an Inverse Function Determining if an Inverse is a Function o One-to-One Functions o Horizontal Line Test Finding the Inverse of a Function Graphs of Inverse Functions Applications

Introduction to Inverse Functions Consider the functions represented by these tables: (a) (b) In tables (c) and (d) that follow, we have swapped the inputs and the outputs from (a) and (b), respectively. (c) (d) When we interchange the input and the output coordinates in a function, we are looking for the inverse of the function. This may result in a relation that is also a function, like examples (a) and (c). However, the new relation may not turn out to be a function, as in the case of (b) and (d); input “9” in (d) has 2 different outputs! Input –3– Output –4–4 979 Input –2– Output –6–6 827 Input –6–6 827 Output –2– Input –4–4 979 Output –3–3 5810

How can we guarantee that the inverse of the original function will also be a function? The original function must be a one-to-one function! A function f is one-to-one if for any elements x 1 and x 2 in its domain, when x 1  x 2 then f(x 1 )  f(x 2 ). That is, any two different inputs will always produce two different outputs. Example: Determine whether the function f(x) = x 3 is one-to-one. Let us check several input-output values. x –3–3 –2–2 –1– f(x) = x 3 – 27 –8–8 –1– Different x-values will produce different y-values. For this function, in general, if x 1  x 2, then (x 1 ) 3  (x 2 ) 3 and we can say that this function is one-to-one.

Determine whether the function f(x) = x is one-to-one. Let x 1 = –2 and x 2 = 2. f(–2) = (–2) = = 9 f(2) = (2) = = 9 Observe that two different inputs will produce the same output. Therefore, the given function is not one-to-one.

Graphical Test: Horizontal Line Test A function is one-to-one if no horizontal line intersects its graph more than once. Revisiting our previous examples: f(x) = x 3 One-to-one function f(x) = x Not one-to-one!

Inverse Functions If f is a one-to-one function with ordered pairs (x, y), the inverse of f, denoted f -1, is also a one-to-one function with ordered pairs (y, x). That is, the inverse of a function is the set of ordered pairs obtained when we swap the inputs and the outputs in the original one-to-one function. The domain of f -1 is the same as the range of f, and the range of f -1 is the same as the domain of f. Note: f -1 is read as "f inverse." Caution: f -1 (x) is the notation for the inverse function and it does not mean the reciprocal of f(x).

Points (–2, 0) and (1, 9) satisfy the function f(x) = 3x + 6. Using this information, show that g (x) is the inverse function of f(x): f(x) = 3x + 6 is a linear function, and we know it is one-to-one, thus, its inverse will also be a function. We know that the inverse of a function is the set of ordered pairs obtained when we swap the inputs and the outputs in the original function. Therefore, we only need to interchange the coordinates of the given points and check that (0, –2) and (9, 1) satisfy g (x). So, g (x) is the inverse function of f(x).

Finding the Inverse of a Function Let f be a one-to-one function defined by y = f(x). (1)Replace f(x) with y. (2) Swap the input and the output (that is, interchange x and y). (3) Solve the new equation for y. (If the equation cannot be solved for y, then the original function has no inverse function.) (4)Let y = f -1 (x). That is, assign the name f -1 (x) to the resulting inverse function.

Find the inverse function of f(x) = 2x – 8. State the domain and range for f(x) and its inverse. f(x) is a linear function and it is one-to-one, thus, its inverse will also be a function. (1) Replace f(x) with y. y = 2x – 8 (2) Interchange x and y. x = 2y – 8 (3) Solve the new equation for y. (continued on the next slide)

(Contd.) (4) Let y = f -1 (x). The domain and range for f(x) is (– ∞, ∞ ), and the domain and range of f - 1 (x) is (– ∞, ∞ ). Optional: We can check that if a point (x, y) satisfies f(x), the swapped coordinates will satisfy the inverse function. Example: (0, –8) satisfies f(x) and (–8, 0) satisfies f - 1 (x).

Find the inverse function of Replace f(x) with y, then interchange x and y. Solve the new equation for y.

Find the inverse of the function: This is a parabola with vertex at (0, –3). As its graph shows next, it is not a one-to-one function (it does not pass the horizontal line test). In this case, we can restrict the domain of f(x) to, let’s say [0, ∞ ), which will guarantee a one-to-one function. (continued on the next slide)

( Contd.) Now we can find the inverse on the limited domain: Observe the domain of f(x) is [0, ∞ ) and it’s range is [–3, ∞ ), while the domain and range of the inverse are [–3, ∞ ) and [0, ∞ ), respectively. (continued on the next slide)

The Graphs of Inverse Functions Recall that if the graph of the original function contains a point (a, b), then the graph of the inverse function will contain the point (b, a). The graph of a point (b, a) is the reflection of the point (a, b) across the line y = x. Thus, we can summarize the following: The graphs of a function and its inverse are symmetric about the line y = x. Example:

Given the graph of f(x), graph its inverse along with y = x; label the inverse g(x). We know that the graphs of a function and its inverse are symmetric about the line y = x. To graph the inverse we only need to reverse the ordered pairs of the original function. The points (2, 4), (1, –3), (0, –4), and (–1, –5) lie on the graph of f(x), therefore, the points (4, 2), (–3, 1), (–4, 0), and (–5, –1) will lie on the graph of the inverse. (continued on the next slide)

(Contd.) We plot the points and connect them with a smooth curve to construct the graph of the inverse, g(x). Observe that both graphs are symmetric about the line y = x.

As of August 1, 2014, $1.00 was equivalent to approximately Euros. a. Write a function f that represents the number of Euros in terms of the number of dollars, x. b. Find the inverse of your function. (continued on the next slide)

(Contd.) c. Find and interpret Round your answer to 2 decimal places. In August 1, 2014, twenty Euros were equivalent to $26.86.

Using your textbook, practice the problems assigned by your instructor to review the concepts from Section 5.1.